Download Constraint Satisfaction Complexity and Logic Phokion G. Kolaitis

Constraint Satisfaction
Complexity
and
Logic
Phokion G. Kolaitis
IBM Almaden Research Center
&
UC Santa Cruz
1
Goals
Goals:
• Show that Constraint Satisfaction Problems (CSP)
form a broad class of algorithmic problems that
arise naturally in several different areas.
• Present an overview of CSP with emphasis on
the computational complexity aspects of CSP
and on the connections with database theory,
univeral algebra, and logic
2
Course Outline
Constraint Satisfaction Problems (CSP)
• Definition & Examples of CSP
• CSP & the Homomorphism Problem
• CSP & Database Theory
• Computational Complexity of CSP
– Uniform & Non-Uniform CSP
– Schaefer’s Dichotomy Theorem for
Boolean CSP
– The Feder-Vardi Dichotomy Conjecture for
general CSP
3
Course Outline
• The Pursuit of Tractable Cases of CSP
– Universal Algebra and Closure Properties
of Constraints
– Datalog, Finite-variable Logics, and Pebble
Games
– Bounded Treewidth and Finite-variable Logics
4
Algorithmic Problems
• k-SAT: (k ≥ 2 is fixed)
Given a k-CNF formula, is it satisfiable?
• k-Colorability: (k ≥ 2 is fixed)
Given a graph H = (V, E), is it k-colorable?
• Clique:
Given a graph H = (V, E) and a positive integer k, does H contain a k-clique?
• Hamiltonian Cycle:
Given a graph H = (V, E), does it contain a
Hamiltonian cycle?
5
Constraint Satisfaction Problems
Problems:
• k-SAT (k ≥ 2)
• k-Colorability (k ≥ 2)
• Clique
• Hamiltonian Cycle
• Latin Square Completion
Question: What do these have in common?
Answer: Each of them is a
Constraint Satisfaction Problem.
6
Constraint Satisfaction Problem
U. Montanari – 1974
CSP – Informal Definition
Input: A set V of variables, a domain D of values
for the variables, and a set C of constraints.
Question: Is there an assignment of values to
variables such that all constraints in C are satisfied?
Example: k-SAT as a CSP
k-CNF ϕ: variables x1 , . . . , xn ; clauses c1 , . . . , cm
• Variables: x1 , . . . , xn
• Values: 0, 1
• Constraints: Given by the clauses of ϕ.
7
Constraint Satisfaction Problem
Input: (V, D, C)
• A finite set V of variables
• A finite domain D of values for the variables
• A set C of constraints (t, R) restricting the
values that tuples of variables can take.
– t:
a tuple t = (x1 , . . . , xm ) of variables
– R: a relation on D of arity |t| = m
Question: Does (V, D, C) have a solution?
Solution: A mapping h : V → D such that
h(t) = (h(x1 ), . . . , h(xm )) ∈ R,
for every constraint (t, R) ∈ C.
An assignment of values to the variables such that
all constraints are satisfied.
8
Example
• 3-Sat:
Given a 3CNF-formula ϕ with variables x1 , . . . , xn
and clauses c1 , . . . , cm , is ϕ satisfiable?
• 3-Sat as a CSP:
– Variables x1 , . . . , xn
– Domain D = {0, 1}
– Constraints ((x, y, z), Ri ), i = 0, 1, 2, 3
Clause
Relation
(x ∨ y ∨ z)
R0 = {0, 1}3 − {(0, 0, 0)}
(¬x ∨ y ∨ z)
R1 = {0, 1}3 − {(1, 0, 0)}
(¬x ∨ ¬y ∨ z)
R2 = {0, 1}3 − {(1, 1, 0)}
(¬x ∨ ¬y ∨ ¬z)
R3 = {0, 1}3 − {(1, 1, 1)}
9
Example
• 3-Colorability:
Given a graph H = (V, E), is it 3-colorable?
• 3-Colorability as a CSP:
– The variables are the nodes in V
– The domain is the set D = {R, G, B} of
three colors.
– The constraints are the pairs ((u, v), Q), where
(u, v) ∈ E and Q is the relation
{(R, G), (G, R), (R, B), (B, R), (B, G), (G, B)}.
10
Constraint Satisfaction Problems
Fundamental & ubiquitous problems in AI and CS:
• Boolean Satisfiability, Graph Colorability, ...
• Database Query Processing
• Planning and Scheduling
• Belief Maintenance
• Machine Vision
• ···
• Linguistics
11
Homomorphisms
Definition: Consider two relational structures
A
B
A = (A, R1A , . . . , Rm
) and B = (B, R1B , . . . , Rm
).
h : A → B is a homomorphism if for every i ≤ m
and every tuple (a1 , . . . , an ) ∈ An ,
RiA (a1 , . . . , an ) =⇒ RiB (h(a1 ), . . . , h(an )).
Example: Clique K4
v
@
v
vs.
v
LQ
QQ
L
Q
Q
L
v
Qv
L
"
Bbb "
L "
B bb
"L
B
b
"
L b
B ""
b L b
"
BB"
bLv
v
v
@
@
@
@
Clique K5
@
@
@v
• There is a homomorphism from K4 to K5 ;
• There is no homomorphism from K5 to K4 .
12
The Homomorphism Problem
Definition: The Homomorphism Problem
Given two relational structures A and B, is there
a homomorphism h : A → B?
Fact: Feder & Vardi – 1993
CSP can be identified with the Homomorphism
Problem.
13
Example
3-Sat as a Homomorphism Problem:
Given 3-CNF formula ϕ with variables x1 , . . . , xn :
• Aϕ = ({x1 , . . . , xn }, R0ϕ , R1ϕ , R2ϕ , R3ϕ ), where
R0ϕ
= {(x, y, z) : (x ∨ y ∨ z) is a clause of ϕ}
R1ϕ
= {(x, y, z) : (¬x ∨ y ∨ z) is a clause of ϕ}
R2ϕ
= {(x, y, z) : (¬x ∨ ¬y ∨ z) is a clause of ϕ}
R3ϕ
= {(x, y, z) : (¬x ∨ ¬y ∨ ¬z) is a clause of ϕ}
• B = ({0, 1}, R0 , R1 , R2 , R3 ), where
R0
= {0, 1}3 − {(0, 0, 0)}
R1
= {0, 1}3 − {(1, 0, 0)}
R2
= {0, 1}3 − {(1, 1, 0)}
R3
= {0, 1}3 − {(1, 1, 1)}
Fact: ϕ is satisfiable iff there is hom. h : Aϕ → B.
14
Examples
3-Colorability as a Homomorphism Problem:
Fact:
A graph H = (V, E) is 3-colorable
⇐⇒
there is a homomorphism h : H → K3 , where K3
is the 3-clique, i.e., K3 = ({R, G, B}, E3 ), where
E3 = {(R, G), (G, R), (R, B), (B, R), (B, G), (G, B)}.
k-Colorability as a Homomorphism Problem:
Fact:
A graph H = (V, E) is k-colorable
⇐⇒
there is a homomorphism h : H → Kk , where Kk
is the k-clique.
15
From CSP to the Homomorphism Problem
Given a CSP-instance (V, D, C), let R1 , . . . , Rm be
the distinct relations on D occurring in C.
Define the instance (A, B) of the Homomorphism
Problem, where
A = (V, {t : (t, R1 ) ∈ C}, . . . , {t : (t, Rm ) ∈ C})
and
B = (D, R1 , . . . , Rm ).
Fact: The CSP instance (V, D, C) has a solution
⇐⇒
there is a homomorphism from A to B.
16
From the Homomorphism Problem to CSP
A
Given an instance A = (A, R1A , . . . , Rm
) and B =
B
(B, R1B , . . . , Rm
) of the Homomorphism Problem, define the instance (V, D, C) of CSP, where
• V = A (i.e., the elements of A are variables);
• D = B (i.e., the elements of B are values);
• C = {(t, RiB ) : t ∈ RiA , 1 ≤ i ≤ m}.
Fact: There is a homomorphism from A to B
⇐⇒
the CSP-instance (V, D, C) has a solution.
17
CSP vs. the Homomorphism Problem
Conclusion: Feder & Vardi – 1993
CSP
= Homomorphism Problem
Some Advantages:
• Clean, algebraic formulation of CSP.
• Makes it possible to apply methods from
universal algebra and logics to the study of
CSP.
• Makes the connections between CSP and database
theory transparent.
18
Course Outline
Constraint Satisfaction Problems
• Definition & Examples of CSP
• CSP & the Homomorphism Problem
• CSP & Database Theory
• Computational Complexity of CSP
• The Pursuit of Tractable Cases of CSP
19
Relational Databases
Definition:
• Relation Schema: R(A1 , . . . , Ak )
R: relation name; A1 , . . . , Ak : attribute names.
• Relation conforming with R:
A set R′ of k-tuples
• Database Schema: A set S of relation schemas
• Database over S: A set S of relations conforming with the relation schemas in S.
Database =
Relational Structure
Example:
• Relation Schemas:
ENROLLS(Student, Course)
TEACHES(Faculty, Course)
• Database Schema:
{ENROLLS, TEACHES}.
20
Relational Databases
Example: (continued)
• Relation Schemas:
ENROLLS(Student, Course)
TEACHES(Faculty, Course)
• Relation ENROLLS
Student
Course
Adams
CS101
Bing
Math10
···
···
• Relation TEACHES
Faculty
Course
Euler
Math10
Turing
CS130
Codd
CS180
···
···
21
Relational Join
Notation: Given relations R1 , . . . , Rm ,
⋉ R2 ⋊
⋉ ··· ⋊
⋉ Rm
R1 ⋊
is the relational join or, simply, the join of R1 , . . . , Rm .
Definition: Given
R1 (A, B, C), R2 (B, C, D), R3 (A, D, E),
R1 ⋊
⋉ R2 ⋊
⋉ R3
= {(a, b, c, d, e) :
R1 (a, b, c) ∧ R2 (b, c, d) ∧ R3 (a, d, e)}
Example: TAUGHT-BY(Student,Course,Faculty)
TAUGHT-BY = ENROLLS ⋊
⋉ TEACHES
Fact: Computing joins is the most frequent and
most costly operation in database systems. (Why?)
22
Relational Joins
• Relation ENROLLS
Student
Course
Adams
CS101
Bing
Math10
···
···
• Relation TEACHES
Faculty
Course
Euler
Math10
Turing
CS130
Codd
CS180
···
···
• Relation TAUGHT-BY
Student
Course
Faculty
Bing
Math10
Euler
···
···
···
23
CSP and Joins
• Consider a CSP instance (V, D, C), where
V
= {X1 , . . . , Xn }
C
= {C1 , . . . , Cm }
Ci
=
(ti , Ri )
ti
=
(Xj1 , . . . , Xji ).
• For each (ti , Ri ), view the occurrence of Ri as a
relation over the relation schema Ri (Xj1 , . . . , Xji ).
Fact: Bibel, Gyssens, Jeavons, ...
R1 ⋊
⋉ ··· ⋊
⋉ Rm = Solutions((V, D, C)).
In particular,
(V, D, C) has a solution ⇐⇒ R1 ⋊
⋉ ··· ⋊
⋉ Rm 6= ∅.
Conclusion: CSP is a database problem.
24
CSP and Databases
Fact: There are strong links between CSP and
key problems in database systems:
• Relational Join Evaluation
• Conjunctive Query Evaluation (CQE)
• Conjunctive Query Containment (CQC)
25
Queries
Definitions:
• Class C of structures: a collection of relational
structures closed under isomorphisms.
• k-ary Query Q on C:
a mapping Q with domain C and such that
– Q(A) is a k-ary relation on A, for A ∈ C;
– Q is preserved under isomorphisms, i.e.,
if h : A → B is an isomorphism, then
Q(B) = h(Q(A)).
• Boolean Query Q on C:
a mapping Q : C → {0, 1} preserved under
isomorphisms. Thus, Q can be identified with
the subclass C ′ of C, where
C ′ = {A ∈ C : Q(A) = 1}.
26
Examples of Queries
• Path of Length 2 – P 2:
Binary query on graphs H = (V, E) such that
P 2(H) = {(a, b) ∈ V 2 : (∃c)(E(a, c) ∧ E(c, b))}.
• Articulation Point – AP :
Unary query on graphs H = (V, E) such that
AP (H) = {a ∈ V : a is an articulation point of H}.
• Connectivity – CN :
Boolean query on graphs H = (V, E) such that

 1 if H is connected
CN (H) =
 0 otherwise.
• k-Colorability, k ≥ 2,
Hamiltonian Cycle, ...
27
Definability of Queries
Let L be a logic and C a class of structures
• A k-ary query Q on C is L-definable if there is
an L-formula ϕ(x1 , . . . , xk ) with x1 , . . . , xk as
free variables and such that for every A ∈ C
Q(A) = {(a1 , . . . , ak ) ∈ Ak : A |= ϕ(a1 , . . . , ak )}.
• A Boolean query Q on C is L-definable if there
is an L-sentence ψ such that for every A ∈ C
Q(A) = 1 ⇐⇒ A |= ψ.
28
Conjunctive Queries
Definition: A conjunctive query is a query definable by a FO-formula in prenex normal form built
from atomic formulas, ∧, and ∃ only.
(∃z1 . . . ∃zm )ψ(x1 , . . . , xk , z1 , . . . , zm ),
where ψ is a conjunction of atomic formulas.
Also, written as a rule:
Q(x1 , . . . , xk ) : − R(y2 , x3 , x1 ), S(x1 , y3 ), . . . , S(y7 , x2 )
Examples:
• Path of Length 2 (Binary query)
(∃z)(E(x1 , z) ∧ E(z, x2 )
P 2(x1 , x2 ) : − E(x1 , z), E(z, x2 )
• Cycle of Length 3 (Boolean query)
(∃x1 ∃x2 ∃x3 )(E(x1 , x2 )∧E(x2 , x3 )∧E(x3 , x1 ))
Q : − E(x1 , x2 ), E(x2 , x3 ), E(x3 , x1 )
29
Conjunctive Queries
• Every relational join is a conjunctive query
R1 (A, B, C), R2 (B, C, D),
R1 ⋊
⋉ R2 (x, y, z, w) : − R1 (x, y, z), R2 (y, z, w)
• Conjunctive Queries are the most frequently
asked queries in databases (a.k.a. SPJ queries)
• The main construct of SQL expresses conjunctive queries
SELECT R1 .A, R2 .D
FROM R1 , R2
WHERE R1 .B = R2 .B AND R1 .C = R2 .C
30
Conjunctive Query Evaluation
A fundamental problem about conjunctive queries
Definition: Conjunctive Query Evaluation CQE
• Given a CQ Q and a structure A, find
Q(A) = {(a1 , . . . ak ) : A |= Q(a1 , . . . , ak )}
• For Boolean queries Q, this becomes:
Given Q and A, does A |= Q? (is Q(A) = 1?)
Examples:
• Given a graph H, find all pairs of nodes connected by a path of length 4.
• Given a graph H, does it contain a triangle?
31
Conjunctive Query Containment
A fundamental problem about conjunctive queries
Definition: Conjunctive Query Containment CQC
• Given two k-ary CQs Q1 and Q2 , is it true that
for every structure A,
Q1 (A) ⊆ Q2 (A)?
• For Boolean queries, this becomes:
Given two Boolean queries Q1 and Q2 , does
Q1 |= Q2 ? (does Q1 logically imply Q2 ?)
Examples:
• Is it true that if two nodes of a graph H are
connected by a path of length 4, then they are
also connected by a path of length 3?
• It is true that if a graph H contains a K4 , then
it also contains a K3 ?
32
CQE, CQC, CSP
Theorem: Chandra & Merlin – 1977
CQE and CQC are the same problem.
Fact: K . . . & Vardi – 1998
CSP and CQC are the same problem
Question: What is the common link?
33
CQE, CQC, CSP
Theorem: Chandra & Merlin – 1977
CQE and CQC are the same problem.
Fact: K . . . & Vardi – 1998
CSP and CQC are the same problem
Question: What is the common link?
Answer: The Homomorphism Problem
34
Canonical CQs and Canonical Structures
Definition: Canonical Conjunctive Query
A
), the canonical CQ of
Given A = (A, R1A , . . . , Rm
A is the Boolean CQ QA with the elements of A
as variables and the “facts” of A as conjuncts:
QA : −
m ^
^
RiA (t)
i=1 t
Definition: Canonical Structure
Given a Boolean conjunctive query Q, let AQ be
the structure with the variables of Q as elements
and the conjuncts of Q as “facts”.
Example:
• A = ({a, b, c}, {(a, b), (b, c), (c, a)}
QA : − E(x, y) ∧ E(y, z) ∧ E(z, x)
• Q : − E(x, y) ∧ E(x, z)
AQ = ({a, b, c), {(a, b), (a, c)}
35
Homomorphisms, CQC and CQE
Theorem: Chandra & Merlin – 1977
For relational structures A and B, TFAE
• There is a homomorphism h : A → B
• B |= QA
(i.e., QA (B) = 1)
• QB ⊆ QA
Theorem: Chandra & Merlin – 1977
For conjunctive queries Q1 and Q2 , TFAE
• Q1 ⊆ Q2
• There is a homomorphism h : AQ2 → AQ1
• AQ1 |= Q2
(i.e., Q2 (AQ1 ) = 1)
36
Homomorphisms, CQC, and CQE
Example: 3-Colorability
For a graph H, the following are equivalent:
1. H is 3-colorable
2. There is a homomorphism h : H → K3
3. K3 |= QH
4. QK3 ⊆ QH
Proof:
(2) =⇒ (3): A homomorphism h : H → K3 provides witnesses in K3 for the existential quantifiers
in QH .
(3) =⇒ (4): If K3 |= QH and A |= QK3 , then
there is a homomorphism h : H → K3 and a homomorphism and h∗ : K3 → A.
The composition h∗ ◦h : H → A provides witnesses
in A for the existential quantifiers in QH .
37
CSP, CQE, and CQC
Theorem: Chandra & Merlin – 1977
For relational structures A and B, TFAE
• There is a homomorphism h : A → B
• B |= QA
(i.e., QA (B) = 1)
• QB ⊆ QA
Conclusion:
CSP, CQE, and CQC are the same problem.
38
Course Outline
Constraint Satisfaction Problems
• Definition & Examples of CSP
• CSP & the Homomorphism Problem
• CSP & Database Theory
• Computational Complexity of CSP
– Uniform & Non-Uniform CSP
– Schaefer’s Dichotomy Theorem for
Boolean CSP
– The Feder-Vardi Dichotomy Conjecture for
general CSP
• The Pursuit of Tractable Cases of CSP
– Datalog & Pebble Games
– Bounded Treewidth and Finite-Variable Logics
39
Computational Complexity Primer
Computational Complexity:
• The quantitative study of solvability
Y. Hartmanis – 1993
• Solvable decision problems are placed into classes
according to the time/space resources required
to solve them.
Some Major Complexity Classes:
P
polynomial time
NP
nondeterministic polynomial time
EXPTIME
exponential time
40
Complexity Classes
Fact: The following containments hold:
P ⊆ NP ⊆ EXPTIME
Theorem: Hartmanis & Stearns – 1965
P ⊂ EXPTIME
Open Problems:
• Is P 6= NP?
• Is NP 6= EXPTIME?
Note: It is generally believed that P 6= NP.
41
NP-Complete Problems
Definition: Let Q be a decision problem (a Boolean
query). We say that Q is NP-complete if
1. Q ∈ NP.
2. Q is NP-hard , i.e., for every Q′ ∈ NP, there
a polynomial-time many-one reduction f of Q′
to Q:
x ∈ Q′ ⇐⇒ f (x) ∈ Q.
Theorem: S. Cook – 1971, R. Karp 1972, ...
The following problems are NP-complete:
• k-Sat, k ≥ 3
• k-Colorability, k ≥ 3
• Clique
• Hamiltonian Cycle
• Latin Square Completion
42
NP-Complete Problems
Fact: Let Q be an NP-complete problem.
The following are equivalent:
• P = NP
• Q ∈ P.
Remark:
• NP-complete problems hold the secret of whether
or not P = NP.
• Establishing that Q is NP-complete is viewed
as evidence that Q 6∈ P and, hence, it is an
intractable algorithmic problem.
43
Complexity of CSP
Definition: Uniform CSP
{(A, B) : ∃ homomorphism h : A → B}
(Homomorphism Problem)
Theorem: Uniform CSP is NP-complete
Proof:
• Uniform CSP is in NP:
Given A, B, “guess” h : A → B and verify it
is a homomorphism.
• Uniform CSP is NP-hard:
It contains NP-hard problems as special cases
(e.g., 3-SAT)
44
Complexity of CQC
Definition: CQC Problem
Given Boolean queries Q1 , Q2 , does Q1 |= Q2 ?
Theorem: CQC is NP-complete
Proof: Same as Uniform CSP
Remark:
• Recall that Boolean CQs are FO-sentences built
from atomic formulas, ∧, and ∃ only.
So, for such sentences, logical implication is
decidable.
• In contrast, for arbitrary FO-sentences, logical
implication is undecidable.
45
Coping with NP-Completeness
Question: How to cope with NP-completeness?
Answer: Garey & Johnson – 1979
• Design heuristic algorithms that may work well
in practice.
• Identify polynomial-time solvable cases of the
problem, the so-called islands of tractability,
by imposing restrictions on the inputs.
Example 1:
– 3-Sat is NP-complete.
– Horn 3-Sat is in P.
Example 2:
– 3-Colorability is NP-complete.
– 3-Colorability on graphs of bounded treewidth
is in P.
46
Restricting Uniform CSP
Research Program: Identify all islands of tractability of Uniform CSP.
Definition: Let A, B be two classes of structures.
CSP(A, B) = {(A, B) : A ∈ A, B ∈ B and
∃ homomorphism h : A → B}
Research Program: Identify all A, B such that
CSP(A, B) is in P.
47
Non-Uniform CSP
Definition: Fix a structure A.
CSP({A}, All) = {B : ∃ homomorphism h : A → B}
Fact: For every fixed A,
CSP({A}, All) ∈ P
Proof: Exercise.
Definition: Fix a structure B:
CSP(All, {B}) = {A : ∃ homomorphism h : A → B}
Fact: The complexity of CSP(All, {B}) depends
on B.
Proof:
• CSP(All, {K3 }) is NP-complete
(3-Colorability)
• CSP(All, {K2 }) is in P.
(2-Colorability)
48
Complexity of Non-Uniform CSP
Notation: For simplicity, for every fixed structure B, we put
CSP(B) = CSP(All, {B})
Research Program:
• Identify the islands of tractability of CSP(B).
• In other words, characterize the structures B
for which CSP(B) is solvable in polynomial
time.
49
Boolean Non-Uniform CSP
Definition: Boolean Non-Uniform CSP
CSP(B) with B = ({0, 1}, R1B , . . . , R1B )
Fact: Boolean Non-Uniform CSP problems are
Generalized Satisfiability Problems, i.e.,
variants of Boolean satisfiability in CNF.
Examples: Each of the following problems is a
CSP(B) problem for a suitable Boolean B.
• 3-SAT: B = ({0, 1}, R0 , R1 , R2 , R3 )
• 2-SAT: B = ({0, 1}, R0′ , R1′ , R2′ )
• POSITIVE 1-IN-3-SAT:
3CNF-formulas with clauses (x ∨ y ∨ z);
exactly one variable is true in each clause.
B = ({0, 1}, R1/3 ),
where
R1/3 = {(1, 0, 0), (0, 1, 0), (0, 0, 1).
50
Complexity of Boolean Non-Uniform CSP
Schaefer’s Dichotomy Theorem – 1978
• If B is Boolean structure, then CSP(B) is in
P or it is NP-complete.
• Moreover, there is a polynomial-time algorithm
to decide, given a Boolean structure B, whether
CSP(B) is in P or it is NP-complete.
Question: Why is this a dichotomy theorem?
51
The Fine Structure of NP
Theorem: Ladner - 1975
If P 6= NP, then there is a problem Q such that
• Q ∈ NP − P;
• Q is not NP-complete.
NP-complete
NP − P
not NP-complete
P
Remark: Consequently, it is conceivable that a
family of NP-problems contains problems of such
intermediate complexity problems.
This possibility cannot be ruled out a priori.
52
Dichotomy of Boolean Non-Uniform CSP
Schaefer’s Dichotomy Theorem – 1978
• If B is Boolean structure, then CSP(B) is in
P or it is NP-complete.
ր
NP-complete
NP − P, not NP-complete
CSP(B)
ց
P
• Moreover, there is a polynomial-time algorithm
to decide, given a Boolean structure B, whether
CSP(B) is in P or it is NP-complete.
Question: What is the boundary in this dichotomy?
53
Dichotomy of Boolean Non-Uniform CSP
Proof Architecture:
B
B = ({0, 1}, R1B , . . . , Rm
)
• Identify a number of cases for which CSP(B)
is in P.
Each such case will be described in terms of
B
conditions on the relations R1B , . . . , Rm
of B.
• Show that CSP(B) is NP-complete in all other
cases.
• Show that there is a polynomial-time algorithm
for checking whether the conditions describing
the tractable cases hold.
54
Tractable Cases of Boolean CSP(B)
Definition: R ⊆ {0, 1}k , for some k ≥ 1.
• R is 0-valid if it contains the k-tuple (0, . . . , 0).
• R is 1-valid if it contains the k-tuple (1, . . . , 1).
B
Fact: B = ({0, 1}, R1B , . . . , Rm
)
• If every RiB is 0-valid, then CSP(B) is in P.
• If every RiB is 1-valid, then CSP(B) is in P.
Proof: Trivial
• The constant function h(x) = 0 is a homomorphism.
• The constant function h(x) = 1 is a homomorphism.
55
Relations and Solutions
Notation: Let ϕ be a Boolean formula with k
variables x1 , . . . , xk .
Sol(ϕ) is the set of all k-tuples (a1 , . . . , ak ) such
that if s is a truth assignment with s(xi ) = ai ,
1 ≤ i ≤ k, then s(ϕ) = 1.
Example: If ϕ is (¬x ∨ y), then
Sol(ϕ) = {(0, 0), (0, 1), (1, 1)}.
Remark: For all remaining tractable cases, the
condition on the relations of B will assert that each
relation RiB is equal to Sol(ϕ), where ϕ belongs to
a certain class of “well-behaved” Boolean formulas
(the class of formulas will change from tractable
case to tractable case).
56
Tractable Cases of Boolean CSP(B)
Definition: R ⊆ {0, 1}k , for some k ≥ 1.
R is bijunctive if there is a 2-CNF formula ϕ such
that R = Sol(ϕ).
B
Theorem: B = ({0, 1}, R1B , . . . , Rm
)
If every RiB is bijunctive, then CSP(B) is in P.
Proof: 2-SAT is in P.
The resolution algorithm runs in polynomial time
on 2-CNF formulas, because
• Every resolvent of two 2-clauses is a 2-clause:
{l1 , l2 }
{l1 , l3 }
ց
ւ
{l2 , l3 }
• Given n variables, there is a polynomial number of 2-clauses involving these variables.
57
Tractable Cases of Boolean CSP(B)
Definition: R ⊆ {0, 1}k , k ≥ 1.
• R is Horn if R = Sol(ϕ) for some Horn formula
ϕ, i.e., a CNF-formula in which each conjunct
has at most one positive literal.
x ∧ (¬x ∨ z) ∧ (¬y ∨ ¬z ∨ w)
• R is dual Horn R = Sol(ϕ) for some dual Horn
formula ϕ, i.e., a CNF-formula in which each
conjunct has at most one negative literal.
¬x ∧ (x ∨ ¬z) ∧ (y ∨ z ∨ ¬w)
B
Theorem: B = ({0, 1}, R1B , . . . , Rm
)
• If every RiB is Horn, then CSP(B) is in P.
• If every RiB is dual Horn, then CSP(B) is in P.
Proof:
Unit Propagation algorithm.
58
Tractable Cases of Boolean CSP(B)
Definition: R ⊆ {0, 1}k , k ≥ 1.
R is affine if it is the set of solutions of a system
of linear equations over the 2-element field.
Example: R = {(0, 1), (1, 0)} is affine, because
R = Sol(x ⊕ y = 1).
B
Theorem: B = ({0, 1}, R1B , . . . , Rm
)
If every RiB is affine, then CSP(B) is in P.
Proof: Use Gaussian Elimination algorithm.
59
Schaefer Structures
B
Definition: B = ({0, 1}, R1B , . . . , Rm
)
B is Schaefer if at least one of the following six
conditions holds:
• Every relation RiB is 0-valid.
• Every relation RiB is 1-valid.
• Every relation RiB is bijunctive.
• Every relation RiB is Horn.
• Every relation RiB is dual Horn.
• Every relation RiB is affine.
Otherwise, we say that B is non-Schaefer.
60
Schaefer’s Dichotomy Theorem
B
Theorem: Let B = ({0, 1}, R1B , . . . , Rm
).
• If B is Schaefer, then CSP(B) is in P.
• If B is non-Schaefer, then CSP(B) is NP-complete.
Proof:
• The first part has already been established.
• The second part requires much more work.
The key ingredient is a theorem about the expressive power of conjunctive queries on nonSchaefer structures.
61
Schaefer’s Expressibility Theorem
Note: Recall that a conjunctive query is a FOformula built from atomic formulas, ∧, and ∃ only.
B
Theorem: Let B = ({0, 1}, R1B , . . . , Rm
).
If B is non-Schaefer, then every Boolean relation
R ⊂ {0, 1}k , k ≥ 1, is definable on B by some
conjunctive query over B with constants 0 and 1.
This means that for every k ≥ 1 and for every
R ⊂ {0, 1}k , there is a formula ψ(x1 , . . . , xk , y, z)
B
that is built from R1B , . . . , Rm
, ∧, and ∃, and has
the property that for every (a1 , . . . , ak ) ∈ {0, 1}k
(a1 , . . . , ak ) ∈ R ⇐⇒ B |= ψ(a1 , . . . , ak , 0, 1).
Remark: Intutitively, this result asserts that on
non-Schaefer structures, conjunctive queries have
maximum expressive power (they can define all
Boolean relations).
62
Schaefer’s Dichotomy Theorem
B
Theorem: Let B = ({0, 1}, R1B , . . . , Rm
).
• If B is Schaefer, then CSP(B) is in P.
• If B is non-Schaefer, then CSP(B) is NP-complete.
• Moreover, there are simple polynomial-time criteria to test if B is Schaefer.
Remark: These criteria involve closure properties
of the relations of B.
63
Closure Properties
Theorem: Let R ⊆ {0, 1}k , k ≥ 1.
• R is bijunctive iff it is closed under the 3-ary
majority function maj(x, y, z).
• S is Horn iff it is closed under g(x, y) = x ∧ y.
• S is dual Horn iff it is is closed under g ′ (x, y) =
x ∨ y.
• S is affine iff it is closed under h(x, y, z) =
x ⊕ y ⊕ z.
64
Closure Properties
Example: Let R = {(0, 1), (1, 0), (1, 1)}
• R is not Horn, because
(0, 1) ∧ (1, 0) = (0, 0) 6∈ R.
• R is not affine, because
(0, 1) ⊕ (1, 0) ⊕ (1, 1) = (0, 0) 6∈ R.
Example: Let R = {(0, 0), (0, 1), (1, 0)}
• R is not dual Horn, because
(0, 1) ∨ (1, 0) = (1, 1) 6∈ R.
• R is not affine, because
(0, 0) ⊕ (0, 1) ⊕ (1, 0) = (1, 1) 6∈ R.
65
Positive 1-in-3 SAT
• 3-CNF formula with clauses (x ∨ y ∨ z);
exactly one variable/clause is true.
• Positive-1-In-3-SAT = CSP(({0, 1}, R1/3 ))
R1/3 = {(1, 0, 0), (0, 1, 0), (0, 0, 1).
Fact: Positive 1-In-3-SAT is NP-complete
Proof: Apply Schaefer’s Dichotomy Theorem
• R1/3 is not bijunctive, because
maj((1, 0, 0), (0, 1, 0), (0, 0, 1)) = (0, 0, 0) 6∈ R.
• R1/3 is neither Horn nor dual Horn, since
(1, 0, 0) ∧ (0, 1, 0) = (0, 0, 0) 6∈ R1/3 .
(1, 0, 0) ∨ (0, 1, 0) = (1, 1, 0) 6∈ R1/3 .
• R1/3 is not affine, since
(1, 0, 0) ⊕ (0, 1, 0) ⊕ (0, 0, 1) = (1, 1, 1) 6∈ R1/3 .
66
Schaefer’s Dichotomy Theorem
B
Theorem: Let B = ({0, 1}, R1B , . . . , Rm
).
• If B is Schaefer, then CSP(B) is in P.
• If B is non-Schaefer, then CSP(B) is NP-complete.
• Moreover, there are simple polynomial-time criteria to test if B is Schaefer.
Remark: Schaefer’s Dichotomy Theorem is the
mother of many other dichotomy results for families of decision problems.
67
Complexity of CSP(B)
Conclusion: Schaefer’s Dichotomy Theorem yields
a complete classification of the complexity of CSP(B)
for Boolean structures B.
Question: What about arbitrary structures B?
68
Complexity of CSP(B)
Conclusion: Schaefer’s Dichotomy Theorem yields
a complete classification of the complexity of CSP(B)
for Boolean structures B.
Question: What about arbitrary structures B?
Dichotomy Conjecture: Feder & Vardi –1993
For every structure B, one of the following holds:
• CSP(B) is in P.
• CSP(B) is NP-complete.
69
Towards the Dichotomy Conjecture
Theorem: Hell & Nešetril – 1990
Let B be an undirected graph.
• If B is 2-Colorable, then CSP(B) is in P.
• If B not 2-Colorable, then CSP(B) is NP-complete.
Theorem: A. Bulatov – 2002
The Dichotomy Conjecture is true for CSP(B),
where B = {0, 1, 2}.
Note: The Dichotomy Conjecture remains open
for CSP(B) with |B| ≥ 4.
70
Islands of Tractability of CSP(B)
Fact: An extensive pursuit of tractable cases of
CSP(B) has taken place over the years.
First Stage: Numerous particular results.
Second Stage: Broad sufficient conditions for
tractability of CSP(B).
Fact: Unifying explanations for many tractability
results have been discovered:
• Expressibility in Datalog
Feder & Vardi – 1993
• Closure Properties of Constraints
Connections with universal algebra
Feder & Vardi – 1993
Jeavons & collaborators – 1995 to present
71
Course Outline
Constraint Satisfaction Problems (CSP)
• Definition & Examples of CSP
• CSP & the Homomorphism Problem
• CSP & Database Theory
• Computational Complexity of CSP
– Uniform & Non-Uniform CSP
– Schaefer’s Dichotomy Theorem for
Boolean CSP
– The Feder-Vardi Dichotomy Conjecture for
general CSP
• The Pursuit of Tractable Cases of CSP
– Datalog & Pebble Games
– Bounded Treewidth & Finite-Variable Logics
72
Datalog
Note: Recall that CQs can be written as rules:
Path of Length 2 - P 2: (∃z)(E(x1 , z) ∧ E(z, x2 )
P 2(x1 , x2 ) : − E(x1 , z), E(z, x2 )
Definition:
• Datalog = Conjunctive Queries + Recursion
Logic Programming without function symbols
• A Datalog program is a finite set of rules given
by conjunctive queries
T (x) : − S1 (y 1 ), . . . , Sr (y r ).
– Some relation symbols may occur both in
the heads and the bodies of rules.
These are the recursive relation symbols or
intensional database predicates (IDBs).
– The remaining relation symbols are the
extensional database predicates (EDBs).
73
Datalog Examples
Definition: Transitive Closure Query T C
Given graph H = (V, E),
T C(H) = {(a, b) ∈ V 2 : there is a path from a to b}.
Example 1: Datalog program for T C
S(x, y) : − E(x, y)
S(x, y) : − E(x, z) ∧ S(z, y)
Example 2: Another Datalog program for T C
S(x, y) : − E(x, y)
S(x, y) : − S(x, z) ∧ S(z, y)
Note:
• E is the EDB.
• S is the IDB; it defines T C.
74
Datalog Examples
Definition: S. Cook – 1974
Path Systems S = (F, A, R)
Given a finite set of formulas F , a set of axioms
A ⊆ F , and a rule of inference R ⊆ F 3 , compute
the theorems of this system.
Example: Datalog program for Path Systems:
T (x) : − A(x)
T (x) : − T (y), T (z), R(x, y, z)
Note:
• A and R are the EDBs.
• T is the IDB; it defines the theorems of S.
75
Tractability of Datalog
Fact:
• If a query Q is defined by a Datalog program,
then Q is in P.
Proof:
• Datalog programs can be evaluated “bottomup” in a polynomial number of iterations.
76
Evaluation of Datalog Programs
Example : Datalog program for T C
S(x, y) : − E(x, y)
S(x, y) : − E(x, z) ∧ S(z, y)
Bottom-up Evaluation
1
S
= ∅
m+1
= {(a, b)) : ∃z(E(a, z) ∧ S m (z, b)}
S
Fact:
S
m
S
m
TC
=
TC
= S |V | .
77
Preservation Properties
Fact: Preservation Properties of Datalog.
• Datalog queries are preserved under
homomorphisms.
• Datalog queries are monotone, i.e., they are
preserved if new tuples are added to the EDBs.
Proof: Exercise - use preservation properties of
positive existential formulas.
Fact: On the class G of all finite graphs:
Datalog[G] $ PTIME[G].
78
Datalog and CSP
B
Fact: Let B = (B, R1B , . . . , Rm
).
• In general, CSP(B) is not monotone. (Why?)
• Hence, CSP(B) is not expressible in Datalog.
However,
• CSP(B) is monotone, where
CSP(B) = {A : 6 ∃ homomorphism h : A → B}.
• Hence, it is conceivable that CSP(B) is
expressible in Datalog (and, thus, it is in P).
79
Datalog and CSP
Fact: Feder & Vardi – 1993
Expressibility of CSP(B) in Datalog is a unifying explanation for many tractability results about
CSP(B).
Example: 2-Colorability = CSP(K2 )
Datalog program for Non 2-Colorability
O(X, Y ) : − E(X, Y )
O(X, Y ) : − O(X, Z), E(Z, W ), E(W, Y )
Q
: − O(X, X)
Example: Let B be Boolean structure.
If B is bijunctive, Horn, or dual Horn, then CSP(B)
is expressible in Datalog.
80
Datalog and CSP
Theorem: Feder & Vardi – 1993
• If B = (B, R1 , . . . , Rk ) is such that
Pol({R1 , . . . , Rk }) contains a near-unanimity
function, then CSP(B) is expressible in Datalog.
• If B = (B, R1 , . . . , Rk ) is such that
Pol({R1 , . . . , Rk }) contains an ACI function,
then CSP(B) is expressible in Datalog.
81
Datalog and CSP
Open Problem: Is there an algorithm to decide
whether, given B, we have that CSP(B) is expressible in Datalog?
B
Question: Fix B = (B, R1B , . . . , Rm
).
When is CSP(B) expressible in Datalog?
82
Datalog and CSP
B
Question: Fix B = (B, R1B , . . . , Rm
).
When is CSP(B) expressible in Datalog?
Answer: K . . . & Vardi – 1998, 2000
Expressibility of CSP(B) in Datalog can be
characterized in terms of
• Finite-Variable Logics and Pebble Games;
• Consistency Properties.
83
The Infinitary Logic Lω
∞ω
Definition: Barwise (1977)
• Infinitary logic with k variables, k ≥ 1.
Lk∞ω is the collection of L∞ω -formulas with at
most k distinct variables.
• Infinitary logic with finitely many variables
[
ω
Lk∞ω .
L∞ω =
k≥1
Fact: Lω
∞ω is a useful tool in analyzing the expressive power of least fixed-point logic.
Theorem: Barwise – 1977, Immerman – 1981
Lω
∞ω -definability can be characterized in terms of
pebble games.
84
Existential Positive Fragments of Lω
∞ω
Definition: K . . . & Vardi – 1995
• ∃Lk∞ω is the fragment of Lk∞ω that contains all
W V
atomic formulas and is closed under ∃, , .
• Existential infinitary logic with finitely many
variables
[
ω
∃Lk∞ω .
∃L∞ω =
k≥1
Fact:
• Datalog ⊆ ∃Lω
∞ω
• ∃Lω
∞ω is a useful tool for analyzing the expressive power of Datalog.
• ∃Lω
∞ω -definability can be characterized in terms
of existential pebble games.
85
Existential k-Pebble Games
Spoiler and Duplicator play on two structures A
and B. Each player uses k pebbles. In each move,
• Spoiler places a pebble on or removes a pebble
from an element of A.
• Duplicator tries to duplicate the move on B.
A:
B:
a1
a2
. . . al
↓
↓
···
↓
b1
b2
...
bl
l≤k
• Spoiler wins the (∃, k)-pebble game if at some
point the mapping ai 7→ bi , 1 ≤ i ≤ l,
is not a partial homomorphism.
• Duplicator wins the (∃, k)-pebble game if the
above never happens.
86
Example
Cliques of Different Size
v
@
v
v
LQ
QQ
L
Q
Q
L
v
Qv
L
b
"
B b " L
"
B bb
"L
B
b
"
L b
B ""
b L b
"
BB"
bLv
v
v
@
@
@
@
@
@
@v
K5
K4
Fact: Let Kk be the k-clique
• Duplicator wins the (∃, k)-pebble game on Kk
and Kk+1 .
• Spoiler wins the (∃, k)-pebble game on Kk and
Kk−1 .
87
Linear Orders of Different Size
w
w
w
w
w
w
w
w
w
w
w
w
w
Lm
Ln
• Spoiler wins the (∃, 2)-pebble game
on Lm and Ln , where m > n.
• Duplicator wins the (∃, 2)-pebble game
on Ln and Lm , where m > n.
88
Winning Strategies in the (∃, k)-Pebble Game
Definition: A winning strategy for the Duplicator
in the (∃, k)-pebble game is a non-empty family I
of partial homomorphisms from A to B such that
• If f ∈ I and h ⊆ f , then h ∈ I
(I is closed under subfunctions).
• If f ∈ I and |f | < k, then for every a ∈ A,
there is g ∈ I so that f ⊆ g and a ∈ dom(g).
(I has the forth property up to k)
89
Uses of Existential k-Pebble Games
• Existential k-pebble games can be used to study
the expressive power of ∃Lk∞ω .
• In particular, existential k-pebble games can
be used to obtain lower bounds for definability
in Datalog.
90
(∃, k)-Pebble Games and ∃Lk∞ω -Definability
Definition: Let A, B be two σ-structures.
A k∞ω B, if every ∃Lk∞ω -sentence that is true on
A is also true on B.
Theorem: K . . . & Vardi – 1995
The following statements are equivalent.
• A k∞ω B.
• Duplicator wins the (∃, k)-pebble game
on A and B.
91
Lower Bounds for ∃Lk∞ω -Definability
Corollary: Let Q be a Boolean query on C. The
following statements are equivalent:
• Q is ∃Lω
∞ω -definable on C.
• There is a k such that for every A, B in C, if
A |= Q and B 6|= Q, then the Spoiler wins the
(∃, k)-pebble game on A, B.
A Methodology for ∃Lω
∞ω -Definability
To show that a Boolean query Q is not expressible
in ∃Lω
∞ω , suffices to show that for every k there are
structures Ak and Bk such that
• Ak |= Q and Bk 6|= Q.
• Duplicator wins the (∃, k)-pebble game
on A, B.
Fact: This method is also complete.
92
k-Datalog
Definition: A k-Datalog program is a Datalog
program in which each rule
t0 : − t1 , . . . , tm
has at most k distinct variables in the head t0 and
at most k distinct variables in the body t1 , . . . , tm .
Example: Non
O(X, Y ) : −
O(X, Y ) : −
Q
:−
2-Colorability revisited
E(X, Y )
O(X, Z), E(Z, W ), E(W, Y )
O(X, X)
Therefore,
Non 2-Colorability is expressible in 4-Datalog.
93
k-Datalog and (∃, k)-Pebble Games
Theorem: K . . . & Vardi
• On the class of all finite structures,
k-Datalog ⊂ ∃Lk∞ω .
• There is a polynomial-time algorithm to decide
whether, given two finite structures A and B,
whether the Spoiler or the Duplicator wins the
(∃, k)-pebble game on A and B.
• For each finite structure B, there is a k-Datalog
program πB that expresses the query: given a
finite σ-structure A, does the Spoiler win the
(∃, k)-pebble game on A and B?
94
Datalog and CSP
Theorem: K . . . & Vardi – 1998
Let k be a positive integer and B a finite structure.
Then the following are equivalent:
• CSP(B) is expressible in k-Datalog
• CSP(B) is expressible in ∃Lk∞ω
• CSP(B) = {A : Duplicator wins the
(∃, k)-pebble game on A and B}.
Note: It is always the case that
CSP(B) ⊆ {A : Duplicator wins the
(∃, k)-pebble game on A and B}.
95
Islands of Tractability of CSP
Non-Uniform CSP(B) Problems
• Definability of CSP(B) in Datalog
• Definability of CSP(B) in ∃Lω
∞ω .
• Single canonical polynomial-time algorithm:
determine who wins the (∃, k)-pebble game.
96
From Non-Uniform CSP to Uniform CSP
Note: So far, we have focused on tractable cases
of non-uniform constraint satisfaction problems.
CSP(B) = CSP(All, {B})
Fact: Many applications in AI and in database
systems involve uniform CSP(A, B), where A and
B are classes of structures.
For example, usually, conjunctive query containment and conjunctive query evaluation appear as
uniform CSPs.
Question:
Can we obtain tractability results for uniform CSP
from tractability results for non-uniform CSP?
97
From Non-Uniform CSP to Uniform CSP
Note: Let A and B be two classes of structures.
It is possible that CSP(A, {B}) is in P, for every
B in B, but CSP(A, B) is NP-complete.
Example:
K = all cliques and G = all graphs:
• CSP(K, {H}) is in P, for every graph H;
• CSP(K, G) in NP-complete.
(it is the Clique problem)
Conclusion: Tractability results for non-uniform
CSP do not necessarily yield tractability results for
uniform CSP.
98
k-Datalog and CSP
However, certain non-uniform tractable cases do
uniformize!
Definition: For every positive integer k, let
Dk
= {B : CSP(B) is expressible in k-Datalog}.
Remark: Recall that if B ∈ Dk , then CSP(B) is
in P.
Theorem:
CSP(All, Dk ) is in P, for every positive integer k.
Proof: An easy consequence of the following:
• Characterization of when CSP(B) is expressible in k-Datalog (using (∃, k)-pebble games).
• Polynomial-time algorithm for determining the
winner in the (∃, k)-pebble game.
99
Datalog and CSP
Conclusions:
• Expressibility in Datalog provides a unifying
explanation for many tractability results about
non-uniform CSP.
• Expressibility of non-uniform CSPs in k-Datalog
uniformizes, which implies that CSP(All, Dk )
is a large island of tractability for uniform CSP.
However,
Open Problem: Let k be a positive integer.
Is there an algorithm to test, given a structure B,
whether or not CSP(B) is expressible in k-Datalog?
100
Bounded Treewidth: Motivation
Observation: Many algorithmic problems that
are “hard” on arbitrary graphs become “easy” on
trees.
Question: Can the concept of tree be weakened
while maintaining “good” computational properties?
101
Bounded Treewidth: Motivation
Observation: Many algorithmic problems that
are “hard” on arbitrary graphs become “easy” on
trees.
Question: Can the concept of tree be weakened
while maintaining “good” computational properties?
Answer: Robertson & Seymour – 1985–1995
Yes, the concept of bounded treewidth achieves this.
Note: Part of their work on graph minors.
(Downey & Fellows: “Parametrized Complexity”)
102
k-Trees, Partial k-Trees, Treewidth
Definition: Let k be a positive integer.
• A graph H is a k-tree if
1. H is a Kk+1 -clique, or
2. There are a k-tree G, nodes a1 , . . . , ak in
G forming a Kk -clique, and a node b in
H − G such that H is obtained from G by
connecting b to each of the nodes a1 , . . . , ak
(thus, forming a Kk+1 -clique).
• A graph is a partial k-tree if it is a subgraph
(not necessarily induced) of a k-tree.
• The treewidth of a graph H, denoted by tw(H),
is the smallest k such that H is a partial k-tree.
103
Treewidth: Examples
Fact:
• If T is a tree, then tw(T) = 1.
• If C is a cycle, then tw(C) = 2.
⋊
⋉) = 2.
• tw(⊟) = 2.
• tw(
• ...
• tw(Kk ) = k − 1, for every k ≥ 2.
104
Treewidth: Characterizations
Fact: The concept of treewidth can be characterized in two different ways:
• Using tree decompositions of graphs.
• Using induced widths of orderings of the nodes.
Note: All three descriptions of treewidth are useful, and are needed in different applications.
105
Treewidth: Complexity
Fact: The following problem is NP-complete:
Given graph H and integer k ≥ 1, is tw(H) ≤ k?
However,
Theorem: Bodlaender – 1993
For every integer k ≥ 1, there is a linear-time algorithm such that, given a graph H, it determines
whether or not tw(H) ≤ k.
106
Bounded Treewidth
Definition: For every integer k ≥ 1,
T (k) = {H : tw(H) < k}.
Fact: Fix an integer k ≥ 2.
Many problems that are NP-complete on arbitrary
graphs become tractable on T (k).
3-Colorability, Monochromatic Triangle,
...
107
Treewidth & Monadic Second-Order Logic
Definition: Monadic Second-Order Logic
(∃X1 ∀X2 · · · ∃Xm )ψ,
where X1 , . . . , Xm are set variables and ψ is firstorder.
Fact: For every m ≥ 1, monadic second-order
logic can express problems that are complete for
the m-th level ΣP
m of the polynomial hierarchy PH.
Theorem: Courcelle – 1990
Let Φ be a sentence of monadic second-order logic.
For every integer k ≥ 2, there is a polynomial-time
algorithm such that, given a graph H ∈ T (k), it
determines whether or not H |= Φ.
108
CSP & Monadic Second-Order Logic
B
Fact: If B = (B, R1B , . . . , Rm
) is a finite structure, then CSP(B) is definable by a sentence of
existential monadic second-order logic with a universal first-order part, i.e., by a sentece of the form
(∃X1 · · · ∃Xn )(∀y1 · · · ∀ys )θ,
where θ is quantifier-free.
Proof: Use one Xi for each element of B.
Example: 3-Colorability
(∃R∃G∃B)(∀y1 ∀y2 )θ.
Corollary: (to Courcelle’s Theorem)
If k ≥ 2, then CSP(T (k), {B}) is in P.
109
Uniform CSP and Bounded Treewidth
Note: The notion of treewidth can be defined for
A
arbitrary finite structures A = (A, R1A , . . . , Rm
).
tw(A) = tw(G(A)),
where G(A) = (A, E) is the Gaifman graph of A,
i.e.,
E(a, a′ ) ⇐⇒ a, a′ occur in a tuple of RiA , for some i.
Theorem: Dechter & Pearl - 1989, Freuder - 1990
If k ≥ 2, then CSP(T (k), All) is in P.
Note: Thus, CSP(T (k), All) is a large island of
tractability for uniform CSP.
Moreover, Bodlaender’s Theorem implies that it is
an easily “accessible” island.
110
Logical Aspects of Bounded Treewidth
Fact:
Fix an integer k ≥ 2.
• The tractability of CSP(T (k), All) can be explained in terms of expressibility in k-Datalog.
• There are tight connections between having
tw(A) < k and the canonical query QA being definable in a fragment of first-order logic
with k variables.
K . . . & Vardi – 1998
Dalmau, K . . ., Vardi – 2002
111
Finite-Variable Logics
Definition: Fix an integer k ≥ 2.
• FOk is the collection of all first-order formulas
with k distinct variables.
• Lk is the collection of all FOk -formulas built
using atomic formulas, ∧, and ∃ only.
Example: Let Cn be the n-element cycle, n ≥ 3.
The canonical conjunctive query QCn is expressible
in L3 .
For instance, QC4 is logically equivalent to
(∃x∃y∃z)(E(x, y)∧E(y, z)∧(∃y)(E(z, y)∧E(y, x))).
It is no accident that tw(Cn ) = 2 ...
112
Bounded Treewidth & Finite-Variable Logics
Theorem: Fix an integer k ≥ 2.
• If A ∈ T (k), then the canonical conjunctive
query QA is expressible in Lk .
• If B is a fixed structure, then CSP(T (k), {B})
is expressible in k-Datalog.
Proof:
• Use the construction of A as a partial k-tree.
• Combine the above with the characterization
of expressibility of non-uniform CSP in k-Datalog
in terms of the (∃, k)-pebble game.
Corollary: CSP(T (k), All) is in P.
Algorithm: Determine whether the Spoiler or
the Duplicator wins the (∃, k)-pebble game.
113
Bounded Treewidth & Finite-Variable Logics
Question: When is QA definable in Lk ?
Definition: A and B are homomorphically equivalent, denoted A ∼h B, if there are homomorphisms h : A → B and h′ : B → A.
Theorem: Fix a k and a finite structure A.
Then the following are equivalent:
• QA is definable in Lk .
• There is some B ∈ T (k) such that A ∼h B.
• core(A) ∈ T (k).
114
Cores
Definition: We say that a structure B is the core
of a structure A if
• B is a submodel of A
• There is no homomorphism h : B → B′ from
B to a proper submodel B′ of B.
Examples:
• core(Kk ) = Kk
• core(Cn ) = Cn
• core(
⋊
⋉) = K
3
• If H is 2-colorable, then core(H) = K2 .
Note: Cores play an important role in database
query processing and optimization.
115
Bounded Treewidth & Finite-Variable Logics
Definition: Fix an integer k ≥ 2.
H(T (k)) is the class of all structures that are
homomorphically equivalent to a structure in T (k).
Remark: T (k) is properly contained in H(T (k)):
There are 2-colorable graphs of arbitrarily large
treewidth (for instance, k-grids)
Theorem : Let k ≥ 2 be an integer.
• If B is a fixed structure, then CSP(H(T (k)), {B})
is definable in k-Datalog.
• CSP(H(T (k)), All) is definable in LFP2k ,
Least Fixed-Point Logic with 2k variables.
Algorithm: Determine whether the Spoiler
or the Duplicator wins the (∃, k)-pebble game.
116
New Islands of Tractability
Theorem: Fix an integer k ≥ 2.
• CSP(H(T (k)), All) is in P.
Good News!
• Determining membership in H(T (k)) is an NPcomplete problem.
Bad News!
Conclusion:
• We have discovered new islands of tractability
that are larger than T (k).
• Unfortunately, these islands are “inaccessible”.
117
Classification Theorem
Question: Are there islands of tractability of the
form CSP(A, All) larger than H(T (k))?
Answer: No!
(modulo a complexity-theoretic assumption)
Theorem: Grohe – 2003
Assume that FPT 6= W [1].
If A is a r.e. class of finite structures over some vocabulary such that CSP(A, All) is in P, then there
is a k ≥ 2 such that A ⊆ H(T (k)).
Note: FPT 6= W [1] is the analog of P 6= NP for
parametrized complexity.
Conclusion: The classes H(T (k)) constitute the
largest islands A of tractability for CSP(A, All).
118
The Archipelago of Tractability for CSP
Classification Program for CSP:
Discover all islands of tractability of CSP:
Identify all classes A and B of finite structures such
that CSP(A, B) is in P.
Current Status:
• All islands of tractability of the form
CSP(A, All)
have been discovered.
• All islands of tractability of the form
CSP(All, {B})
with |B| ≤ 3 have been discovered.
• Much more remains to be done. So, the quest
goes on ...
119
Bounded Treewidth & Finite-Variable Logics
Question: Is there a more precise relationship
between T (k) and Lk ?
Theorem: Fix a k and a finite structure A.
Then the following are equivalent:
• A ∈ T (k)
• QA can be transformed to an Lk -sentence by
repeatedly applying the following rewrite rules
to subformulas of QA :
– Associativity and Commutativity of ∧
– (∃x)(ϕ ∧ ψ) 7→ (∃x)ϕ ∧ ψ, provided x is
not free in ψ.
– (∃x)ϕ 7→ (∃y)ϕ[x/y], provided y is not
free in θ.
Note: The above rewrite rules are routinely used
in database query processing and optimization.
120
Concluding Remarks
Constraint Satisfaction is a meeting point of
• Computational Complexity
• Database Theory
• Logic
• Universal Algebra.
121